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Jun 29, 2018
06/18

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Lisette Jager

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The construction, in [AJN], of a pseudodifferential calculus analogous to the Weyl calculus, in an infinite dimensional setting, required the introduction of convenient classes of symbols. In this article, we proceed with the study of these classes in order to establish, later on, the properties that a pseudodifferential calculus is expected to satisfy. The introduction and the study of a new class are rendered necessary in view of applications in QED. We prove here that the symbols of both...

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1607.02253

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Jun 29, 2018
06/18

by
Simon Jäger; Dieter Rautenbach

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For a set $S$ of vertices of a graph $G$, a vertex $u$ in $V(G)\setminus S$, and a vertex $v$ in $S$, let ${\rm dist}_{(G,S)}(u,v)$ be the distance of $u$ and $v$ in the graph $G-(S\setminus \{ v\})$. Dankelmann et al. (Domination with exponential decay, Discrete Math. 309 (2009) 5877-5883) define $S$ to be an exponential dominating set of $G$ if $w_{(G,S)}(u)\geq 1$ for every vertex $u$ in $V(G)\setminus S$, where $w_{(G,S)}(u)=\sum\limits_{v\in S}\left(\frac{1}{2}\right)^{{\rm...

Topics: Combinatorics, Mathematics

Source: http://arxiv.org/abs/1605.05991

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Jun 27, 2018
06/18

by
Jing Wang; Tobias Jäger

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We study the phenomenon of mode-locking in the context of quasiperiodically forced non-linear circle maps. As a main result, we show that under certain C1-open condition on the geometry of twist parameter families of such systems, the closure of the union of modelocking plateaus has positive measure. In particular, this implies the existence of infinitely many mode-locking plateaus (open Arnold tongues). The proof builds on multiscale analysis and parameter exclusion methods in the spirit of...

Topics: Mathematics, Dynamical Systems

Source: http://arxiv.org/abs/1504.03887

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2.0

Jun 28, 2018
06/18

by
M. Gröger; T. Jäger

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The aim of this note is to point out some observations concerning modified power entropy of $\Z$- and $\N$-actions. First, we provide an elementary example showing that this quantity is sensitive to transient dynamics, and therefore does not satisfy a variational principle. Further, we show that modified power entropy is not suitable to detect the break of equicontinuity which takes place during the transition from almost periodic to almost automorphic minimal systems. In this respect, it...

Topics: Dynamical Systems, Mathematics

Source: http://arxiv.org/abs/1506.07192

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Jun 27, 2018
06/18

by
Tobias Jäger; Andres Koropecki

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We extend Poincar\'e's theory of orientation-preserving homeomorphisms from the circle to circloids with decomposable boundary. As special cases, this includes both decomposable cofrontiers and decomposable cobasin boundaries. More precisely, we show that if the rotation number on an invariant circloid $A$ of a surface homeomorphism is irrational and the boundary of $A$ is decomposable, then the dynamics are monotonically semiconjugate to the respective irrational rotation. This complements...

Topics: Dynamical Systems, Mathematics

Source: http://arxiv.org/abs/1506.01096

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Jun 30, 2018
06/18

by
T. Jäger; F. A. Tal

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We provide an equivalent characterisation for the existence of one-dimensional irrational rotation factors of conservative torus homeomorphisms that are not eventually annular. It states that an area-preserving non-annular torus homeomorphism $f$ is semiconjugate to an irrational rotation $R_\alpha$ of the circle if and only if there exists a well-defined speed of rotation in some rational direction on the torus, and the deviations from the constant rotation in this direction are uniformly...

Topics: Mathematics, Dynamical Systems

Source: http://arxiv.org/abs/1410.3662

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5.0

Jun 29, 2018
06/18

by
Michael A. Henning; Simon Jäger; Dieter Rautenbach

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The domination number $\gamma(G)$ of a graph $G$, its exponential domination number $\gamma_e(G)$, and its porous exponential domination number $\gamma_e^*(G)$ satisfy $\gamma_e^*(G)\leq \gamma_e(G)\leq \gamma(G)$. We contribute results about the gaps in these inequalities as well as the graphs for which some of the inequalities hold with equality. Relaxing the natural integer linear program whose optimum value is $\gamma_e^*(G)$, we are led to the definition of the fractional porous...

Topics: Combinatorics, Mathematics

Source: http://arxiv.org/abs/1605.04575

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Jun 29, 2018
06/18

by
Michael A. Henning; Simon Jäger; Dieter Rautenbach

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We characterize a large subclass of the class of those graphs $G$ for which the exponential domination number of $H$ equals the domination number of $H$ for every induced subgraph $H$ of $G$.

Topics: Combinatorics, Mathematics

Source: http://arxiv.org/abs/1605.05056

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1.0

Jun 30, 2018
06/18

by
Katja Polotzek; Kathrin Padberg-Gehle; Tobias Jäger

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We establish a set-oriented algorithm for the numerical approximation of the rotation set of homeomorphisms of the two-torus homotopic to the identity. A theoretical background is given by the concept of {\epsilon}-rotation sets. These are obtained by replacing orbits with {\epsilon}-pseudo-orbits in the definition of the Misiurewicz-Ziemian rotation set and are shown to converge to the latter as {\epsilon} decreases to zero. Based on this result, we prove the convergence of the numerical...

Topics: Dynamical Systems, Mathematics

Source: http://arxiv.org/abs/1702.06190

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Jun 30, 2018
06/18

by
Laurent Amour; Lisette Jager; Jean Nourrigat

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In this paper we establish explicit lower bounds for pseudodifferential operators with a radial symbol. The proofs use classical Weyl calculus techniques and some useful, if not celebrated, properties of the Laguerre polynomials.

Topics: Mathematics, Analysis of PDEs

Source: http://arxiv.org/abs/1403.7368

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Jun 30, 2018
06/18

by
Lisette Jager; Jules Maes; Alain Ninet

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We study the real valued process $ \{X_t, t\in {\mathbb N}\} $ defined by $X_{t+2} = \varphi(X_t,X_{t+1})$, where the $X_t$ are bounded. We aim at proving the decay of correlations for this model, under regularity assumptions on the transformation $\varphi$.

Topics: Mathematics, Dynamical Systems

Source: http://arxiv.org/abs/1412.2644

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Jun 29, 2018
06/18

by
Jing Wang; Qi Zhou; Tobias Jäger

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We show that a generic quasiperiodically forced circle homeomorphism is mode-locked: the rotation number in the fibres is rationally related to the rotation number in the base and it is stable under small perturbations of the system. As a consequence, this implies that for a generic parameter family of quasiperiodically forced circle homeomorphisms satisfying a twist condition, the graph of the rotation number as a function of the parameter is a devil's staircase.

Topics: Dynamical Systems, Mathematics

Source: http://arxiv.org/abs/1607.01700

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Jun 30, 2018
06/18

by
Tobias Jäger; Alejandro Passeggi; Sonja Štimac

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We study the rotational behaviour on minimal sets of torus homeomorphisms and show that the associated rotation sets can be any type of line segments as well as non-convex and even plane-separating continua. This shows that restrictions holding for rotation sets on the whole torus are not valid on minimal sets. The proof uses a construction of rotational horseshoes by Kwapisz to transfer the problem to a symbolic level, where the desired rotational behaviour is implemented by means of suitable...

Topics: Mathematics, Dynamical Systems

Source: http://arxiv.org/abs/1408.2931

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Jun 30, 2018
06/18

by
Lisette Jager; Jules Maes; Alain Ninet

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We study the discrete-time, real valued bounded process $\{X_n, n\in {\mathbb N} \}$ defined by a second order recurrence relation $X_{n+2} = \varphi(X_n,X_{n+1})$. We obtain the decay of correlations under analytical hypotheses on $\varphi $ (regularity, bounds on derivatives ...)

Topics: Mathematics, Dynamical Systems

Source: http://arxiv.org/abs/1411.0453

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Jun 30, 2018
06/18

by
Laurent Amour; Lisette Jager; Jean Nourrigat

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We aim at extending the definition of the Weyl calculus to an infinite dimensional setting, by replacing the phase space $ \mathbb{R}^{2n}$ by $B^2$, where $(i,H,B)$ is an abstract Wiener space. A first approach is to generalize the integral definition using the Wigner function. The symbol is then a function defined on $B^2$ and belonging to a $L^1$ space for a gaussian measure, the Weyl operator is defined as a quadratic form on a dense subspace of $L^2(B)$. For example, the symbol can be the...

Topics: Mathematics, Analysis of PDEs

Source: http://arxiv.org/abs/1412.1577

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Jun 30, 2018
06/18

by
Gabriel Fuhrmann; Maik Gröger; Tobias Jäger

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We study the geometric and topological properties of strange non-chaotic attractors created in non-smooth saddle-node bifurcations of quasiperiodically forced interval maps. By interpreting the attractors as limit objects of the iterates of a continuous curve and controlling the geometry of the latter, we determine their Hausdorff and box-counting dimension and show that these take distinct values. Moreover, the same approach allows to describe the topological structure of the attractors and to...

Topics: Mathematics, Dynamical Systems

Source: http://arxiv.org/abs/1412.6054

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Jun 29, 2018
06/18

by
Lisette Jager; Jules Maes; Alain Ninet

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We study the real, bounded-variables process (X_n) defined by a k-term recurrence relation X_{n+k} ={\phi}(X_n, ... , X_{n+k-1}). We prove the decay of correlations, mainly under purely analytic hypotheses concerning the function {\phi} and its partial derivatives.

Topics: Dynamical Systems, Mathematics

Source: http://arxiv.org/abs/1606.01644

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Jun 29, 2018
06/18

by
Tobias Jäger; Daniel Lenz; Christian Oertel

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We construct model sets arising from cut and project schemes in Euclidean spaces whose associated Delone dynamical systems have positive toplogical entropy. The construction works both with windows that are proper and with windows that have empty interior. In a probabilistic construction, the entropy almost surely turns out to be proportional to the measure of the boundary of the window.

Topics: Dynamical Systems, Mathematics

Source: http://arxiv.org/abs/1605.01167

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3.0

Jun 27, 2018
06/18

by
G. Fuhrmann; M. Gröger; T. Jäger

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We introduce amorphic complexity as a new topological invariant that measures the complexity of dynamical systems in the regime of zero entropy. Its main purpose is to detect the very onset of disorder in the asymptotic behaviour. For instance, it gives positive value to Denjoy examples on the circle and Sturmian subshifts, while being zero for all isometries and Morse-Smale systems. After discussing basic properties and examples, we show that amorphic complexity and the underlying asymptotic...

Topics: Mathematics, Dynamical Systems

Source: http://arxiv.org/abs/1503.01036

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2.0

Jun 28, 2018
06/18

by
François Béguin; Sylvain Crovisier; Tobias Jäger

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We build an irrational pseudo-rotation of the 2-torus which is semiconjugate to an irrational rotation of the circle in such a way that all the fibres of the semi-conjugacy are pseudo-circles. The proof uses the well-known `fast-approximation method' introduced by Anosov and Katok.

Topics: Dynamical Systems, Mathematics, General Topology

Source: http://arxiv.org/abs/1508.07022

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Jun 29, 2018
06/18

by
Michael Gentner; Irene Heinrich; Simon Jäger; Dieter Rautenbach

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A prominent parameter in the context of network analysis, originally proposed by Watts and Strogatz (Collective dynamics of `small-world' networks, Nature 393 (1998) 440-442), is the clustering coefficient of a graph $G$. It is defined as the arithmetic mean of the clustering coefficients of its vertices, where the clustering coefficient of a vertex $u$ of $G$ is the relative density $m(G[N_G(u)])/{d_G(u)\choose 2}$ of its neighborhood if $d_G(u)$ is at least $2$, and $0$ otherwise. It is...

Topics: Combinatorics, Mathematics

Source: http://arxiv.org/abs/1611.06135

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Jun 30, 2018
06/18

by
Lorenzo J. Díaz; Katrin Gelfert; Maik Gröger; Tobias Jäger

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We study fractal properties of invariant graphs of hyperbolic and partially hyperbolic skew product diffeomorphisms in dimension three. We describe the critical (either Lipschitz or at all scales H\"older continuous) regularity of such graphs. We provide a formula for their box dimension given in terms of appropriate pressure functions. We distinguish three scenarios according to the base dynamics: Anosov, one-dimensional attractor, or Cantor set. A key ingredient for the dimension...

Topics: Dynamical Systems, Mathematics

Source: http://arxiv.org/abs/1702.06416